雙曲幾何講義(影印版)（簡體書）

作者：（意大利）本尼迪特（Riccardo Benedetti） （意大利）Carlo Petronio

《雙曲幾何講義(英文版)》由世界圖書出版公司出版。

Preface
Chapter A
Hyperbolic Space
A.1 Models for Hyperbolic Space
A.2 Isometrics of Hyperbolic Space:Hyperboloid Model
A.3 Conformal Geometry
A.4 Isometrics of Hyperbolic Space:Disc and Half-space Models
A.5 Geodesics, Hyperbolic Subspaces and Miscellaneous Facts
A.6 Curvature of Hyperbolic Space
Chapter B.
Hyperbolic Manifolds
and the Compact Two-dimensional Case
B.1 Hyperbolic, Elliptic and Flat Manifolds
B.2 Topology of Compact Oriented Surfaces
B.3 Hyperbolic, Elliptic and Flat Surfaces
B.4 Teichmüller Space
Chapter C.
The Rigidity Theorem (Compact Case)
C.1 First Step of the Proof:Extension of Pseudo-isometrics
C.2 Second Step of the Proof:Volume of Ideal Simplices
C.3 Gromov Norm of a Compact Manifold
C.4 Third Step of the Proof:
the Gromov Norm and the Volume Are Proportional
C.5 Conclusion of the Proof, Corollaries and Generalizations
Chapter D.
Margulis' Lemma and its Applications
D.1 Margulis' Lemma
D.2 Local Geometry of a Hyperbolic Manifohl
D.3 Ends of a Hyperbolic Manifold
Chapter E.
The Space of Hyperbolic Manifolds
and the Volume Function
E.1 The Chabauty anti the Geometric Topology
E.2 Convergence in the Geometric Topology:Opening Cusps.The Case of Dimension at least Three
E.3 The Case of Dimension Different from Three.Conclusions and Examples
E.4 The Three-dimensional Case:Jorgensen's Part of the So-called Jorgensen-Thurston Theory
E.5 The Three-dimensional Case. Thurston's Hyperbolic Surgery Theorem:Statement and Preliminaries
E.5-ⅰ Definition and First Properties of T3 (Non-compact Three-manifolds with "Triangulation" Without Vertices)
E.5-ⅱ Hyperbolic Structures on an Element of T3 and Realization of the Complete Structure
E.5-ⅲ Elements of T3 and Standard Spines
E.5-ⅳ Some Links Whose Complementsare Realized as Elements of T3
E.6 Proof of Thurston's Hyperbolic Surgery Theorem
E.6-ⅰ Algebraic Equations of H(N1)(Hyperbolic Structures Supported by M ∈T3)
E.6-ⅱ Dimension of H(M):General Case
E.6-ⅲ The Case M is Complete Hyperbolic:the Space of Deformations
E.6-ⅳ Completion of the Deformed Hyperbolic Structetures and Conclusion of the Proof
E.7 Applications to the Study of tim Volume Fnnction and Complements about Three-dimensional Hyperbolic Geometry
Chapter F.
Bounded Cohomology, a Rough Outline
F.1 Singular Cohomology
F.2 Bounded Singular Cohomology
F.3 Flat Fiber Bundles
F.4 Euler Class of a Flat Vector Bundle
F.5 Flat Vector Bundles on Surfaces and the Milnor-Sullivau Theorem
F.6 Sullivan's Conjecture and Amenable Groups
Subject Index
Notation Index
References

We are going to see that these contributions cancel each other; we shall dothis by considering four differerent possibilities for a triple v, E, F and show-ing that to each contribution +1 there corresponds an opposite contributiondue to a triple v', E', F' being in the same case; the verification that thesecontributions actually cancel each other in pairs will then be obvious.
Given v, E, F let us consider as preferred endpoint of E the one corre-sponding to the torus L. The four cases we axe going to consider correspondto the possible answers to the following questions:
(A) is the vertez common to E and F the preferred endpoint of E?
(B) does the face of Tv containing E and F abo contain an edge of α?
(From now on the proof will make extcnsive use of pictures; we are goingto realize all the tetrahedra in Ⅲ31+ ; as for major evidence, in the situation inexam, we are going to give E the orientation towards its preferred endpoint:this is not so useful now, as only E has a preferred endpoint, but it will bevery useful later when proving (1)'. In the present situation we are going torealize always the preferred endpoint of E as ∞.)
(A) Yes. (B) Yes.
If σ(v, E, F) = -1 the situation is the one represented in Fig. E.47. Thenwe can consider the edge of α coming before the one lying on the face of Tvcontaining E and F. Let Tv1, be the tetrahedron corresponding to the trianglelying on the right, of this edge of α; with the choice of F' = F and E' as inFig. E.48 we obviously have that the triple v', E', F' contributes to the sumtoo and σ(v', E', F') = +1.